## Using Mathematical Models to Assess Responses to an Outbreak of an Emerged Viral Respiratory Disease

We now assess the effect of interventions involving the use of antiviral drugs by a different measure, namely with reference to *q*, the probability that an outbreak initiated by a single infected individual fades out before gathering momentum. Recall that the information that *R* contains about *q* is that R < 1 implies *q *= 1. For *R* > 1 we are interested in the value of *q*, corresponding to different interventions.

In Figure 5.3 we show the value of *q* for a range of *R*_{0} values corresponding to different interventions. The model and the interventions considered are as in Section 5.2, for the SEIR model with three types of individual (an individual is either a GP, a HCW or a member of the general public).

**Figure 5.3** The probability *q* that an outbreak initiated by a single infected individual fails to gather momentum as a function of *R*_{0}, for a variety of targeted antiviral coverage.

**(a)** *ƒ* =0.8, i.e. late isolation of cases, and

**(b) ***ƒ* =0.5, i.e. timely isolation of cases.

The curves in Figure 5.3 are consistent with the results seen in Figure 5.2, in terms of the magnitude that each additional component adds to the effectiveness of the combined intervention. Specifically, if in addition to using the default AV strategy for health care workers it is possible to protect 65% of exposed individuals, by targeted prophylactic use of AVs, one is likely to eliminate the infection.

As the targeted use of antiviral drugs for prophylaxis seems effective it is instructive to show the result over the entire range of coverage that might be achieved. This is done in Figure 5.4, where the horizontal axis is *g*, the proportion of exposed individuals who are administered AVs for prophylaxis. (Figure 5.2 and Figure 5.3 show results only for *g* = 0.2, 0.4 and 0.65.) Figure 5.4 illustrates directly the impact of varying *g* (the proportion of the population successfully targeted) for a low (*R*_{0}=1.5), medium (*R*_{0}=2.5) and high (*R*_{0}=3.5) value of *R _{0}*. The results demonstrate that relatively low values of

*g*are adequate for significant increases in the probability of containment. For example, when

*R*=2.5 and

_{0}*f*=0.8, targeting 50% of the population will achieve elimination.

**Figure 5.4** The probability *q* that the infection is eliminated before it establishes itself as a function of *g*, the proportion of the exposed individuals receiving AVs for prophylaxis, for *R*_{0}= 1.5, 2.5 and 3.5.

**(a)** *ƒ* =0.8, i.e. late isolation of cases, and **(b) ***ƒ* =0.5, i.e. timely isolation of cases.

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